Quantum cluster variables via vanishing cycles

TL;DR

This paper offers a Hodge-theoretic perspective on quantum cluster algebras, linking positivity to monodromic mixed Hodge structures and proving positivity under certain acyclicity conditions.

Contribution

It introduces a novel Hodge-theoretic interpretation of Laurent phenomenon in quantum cluster algebras using Donaldson-Thomas theory.

Findings

Positivity conjecture reduces to purity of monodromic mixed Hodge structures.

Positivity and Lefschetz properties hold if initial or mutated seed is acyclic.

Acyclic initial seed positivity previously established by Qin and Nakajima.

Abstract

In this paper, we provide a Hodge-theoretic interpretation of Laurent phenomenon for general skew-symmetric quantum cluster algebras, using Donaldson-Thomas theory for a quiver with potential. It turns out that the positivity conjecture reduces to the certain statement on purity of monodromic mixed Hodge structures on the cohomology with the coefficients in the sheaf of vanishing cycles on the moduli of stable framed representations. As an application, we show that the positivity conjecture (and actually a stronger result on Lefschetz property) holds if either initial or mutated quantum seed is acyclic. For acyclic initial seed the positivity has been already shown by F. Qin \cite{Q} in the quantum case, and also by Nakajima \cite{Nak} in the commutative case.